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Proposal for a new quantum theory of gravity III: Equations for quantum gravity, and the origin of spontaneous localisation

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Pith's one-line read A single trace action for an atom of space-time-matter yields quantum gravity and, after spontaneous localisation, Einstein's equations.

desk verdict A bold two-beta trace action for an STM atom that is meant to unify gravity, quantum theory, and spontaneous localisation, but whose central derivation rests on an undefended scale relation presented as a prediction. read the letter →

arxiv 1908.04309 v2 pith:BT7YQSX4 submitted 2019-08-12 gr-qc hep-thquant-ph

Palemkota Maithresh , Tejinder P. Singh This is my paper · ORCID
classification gr-qchep-thquant-ph
keywords quantumgravityspontaneouslocalisationnon-commutativegeometrySTMatomDiracoperatorConnestimeemergentspacetimegeneralrelativity
topics Quantum Gravity
open problems Quantum Gravity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proposes a quantum theory of gravity built from a single trace action for an 'atom of space-time-matter' (STM atom). The central claim is that this one action contains both gravity and matter, that its anti-self-adjoint part produces spontaneous localisation without being added by hand, and that localisation converts the quantum dynamics into classical general relativity with point-particle matter sources. A sympathetic reader should care because the theory claims to explain the quantum-to-classical transition and the origin of classical spacetime from the same principle that gives Einstein's equations.

What carries the argument

The central object is the STM atom's trace action $S = \frac{L_P}{C_0} \frac{1}{2} \int d\tau \, \mathrm{Tr}\left[(\dot q_B + \beta_1 \dot q_F)(\dot q_B + \beta_2 \dot q_F)\right]$, where $\tau$ is Connes time, $q_B$ governs gravity through the Dirac operator $D_B$, and $q_F$ is fermionic matter. The mechanism that carries the argument is the split of the total operator $D = D_B + D_F$ and of its eigenvalues into real and imaginary parts, $\lambda_R = 1/L$ and $\lambda_I = 1/L_I$; the assumption $L_P^2 L_I = L^3$ turns localisation of a fermion into a point-particle source, while the complex length $L_{\mathrm{com}} = L + i R_S$ interpolates between Dirac fermion and black hole limits. This machinery converts a free matrix dynamics into quantum gravity, then into classical general relativity.

What would settle it

An experiment that measures the spontaneous localisation rate of a single nucleon and finds it significantly different from $L_I/c \sim 10^{-17}\,\mathrm{s}^{-1}$ (with $L_I \approx 10^{27}\,\mathrm{cm}$) would falsify the central derivation; a null result for collapse in coordinate time would also falsify the relativistic localisation claim.

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Extended reading notes

Core claim

The paper claims that the equations of quantum gravity follow from a single trace Lagrangian for the STM atom, with the operator $q = q_B + q_F$ split into bosonic and fermionic parts and two constant fermionic matrices $\beta_1, \beta_2$ in the action. After statistical thermodynamics of an ensemble of these atoms, quantum commutation relations emerge and the self-adjoint part of the Hamiltonian gives a Schrödinger equation in Connes time; the anti-self-adjoint part drives spontaneous localisation of fermionic degrees of freedom, which defines an emergent classical spacetime. Once localisation happens, the trace action reduces to the classical action $S = \int d^4 x \, \sqrt{g} \left[ \frac{c^3}{2G} R + c \sum_i m_i \delta^3(x-x_0) \right]$, and the eigenvalue equation $(D_B + D_F)\psi = \frac{1}{L}\left(1 + i \frac{L_P^2}{L^2}\right)\psi$ is claimed to unify the Dirac equation with Einstein equations, with fermions and black holes as opposite limits.

Load-bearing premise

The derivation's load-bearing premise is that spontaneous localisation confines an STM atom to a spatial volume $L^3$, so that the product $L_P^2 L_I$ equals $L^3$; if that relation is wrong, the recovered matter action and the predicted collapse rate and proton mass all fail.

Editorial extensions

If this is right

  • General relativity with matter is not quantised but emerges as the commutative limit of a non-commutative matrix dynamics; the gravitational field is an emergent condensate.
  • Spontaneous localisation has a dynamical origin in the anti-self-adjoint part of the Hamiltonian, so collapse is intrinsic to quantum gravity rather than postulated.
  • The theory predicts a collapse rate of roughly $L_I/c \sim 10^{-17}\,\mathrm{s}^{-1}$ for a nucleon, matching the rates used in standard localisation models, and an amplified rate $N \times 10^{-17}\,\mathrm{s}^{-1}$ for a body of $N$ nucleons.
  • Equation (68) makes black holes and Dirac fermions two limits of one eigenvalue equation, explaining why a Kerr-Newman black hole has the electron's gyromagnetic ratio.
  • The relation $L_I = L^3/L_P^2$ ties the proton mass to the Hubble scale, $m_{\mathrm{pr}}/m_P \approx \left( L_P/(c H_0^{-1}) \right)^{1/3}$, linking particle masses to cosmology.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the relation $L_I = L^3/L_P^2$ holds, then the same length scale sets the collapse rate, the proton mass, and the size of the observable universe, so the theory converts a coincidence into a testable numerical chain.
  • Because localisation is seeded by the fermionic part of the Dirac operator, relativistic collapse should localise coordinate time as well as position; experiments that probe collapse in time could distinguish this theory from non-relativistic collapse models.
  • The black hole/fermion duality $L \leftrightarrow i L_P^2/L$ suggests that black hole entropy could be computed from the spectrum of the Dirac operator on the dual fermion side; the paper hints at but does not prove this.
  • The two beta matrices entering the action naturally suggest a two-dimensional structure at the Planck scale; if so, the STM atom may be a precursor of string- or loop-like degrees of freedom (the paper itself raises this possibility).
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper proposes a 'Non-commutative Matter-Gravity' theory by extending the authors' earlier work. Its starting point is a trace-dynamics action for an STM atom, Eq. (7), whose bosonic and fermionic parts are described by qB and qF and two constant fermionic matrices β1 and β2. The paper derives the Lagrange and Hamilton equations, constructs the trace Hamiltonian and the Adler-Millard charge, and argues that the anti-self-adjoint part of the Hamiltonian is the seed of spontaneous localisation. It then asserts that after statistical thermodynamics (postponed to a forthcoming paper) the system acquires the quantum commutation relations (53), and that after spontaneous localisation the trace action reduces to the Einstein-Hilbert action with point-particle matter sources, Eq. (64). The final part identifies the imaginary part of the Dirac-operator eigenvalue λI = 1/L_I with L_I = L^3/L_P^2, which is used to propose a collapse rate ∼10^{-17} s^{-1}, a Dirac/black-hole unification, and a proton-mass formula (71).

Significance. If established, the theory would be highly significant: it would show that a single trace action can generate quantum commutation relations, spontaneous wave-function collapse, and classical general relativity with matter, while making concrete numerical predictions. The paper contains a coherent derivation of the Level-0 free-particle dynamics, and it is commendably explicit about open items such as the operator/metric relation, Lorentzian continuation, and higher-order heat-kernel terms. No machine-checked proofs or reproducible code are provided, and the central claims rest on assumptions that are not derived within the manuscript. The significance is therefore programmatic rather than demonstrative.

major comments (4)
  1. [Section II, between Eqs. (62) and (63)] The derivation of the classical matter action from the trace action is incomplete. The term (62) is converted into the point-particle action (63) only by the assumption L_p^2 L_I = L^3, introduced in the sentence 'We make the assumption ... that spontaneous localisation localises the STM atom to a spatial volume L^3 such that L_p^2 L_I = L^3.' The text says this will 'become plausible shortly', but the plausibility argument that follows is just the same relation used to estimate L_I from a nucleon Compton wavelength. Since Eq. (64) — the central claim that classical general relativity with matter is recovered — depends on this step, the relation is load-bearing and not a cosmetic ansatz. Section III's list of outstanding issues does not flag this assumption as unresolved.
  2. [Section II, Eq. (53)] The quantum commutation relations (53), and the Ward identity from which they are said to follow, are not derived in this manuscript. The paper states that the statistical thermodynamics 'will be described in detail in a forthcoming work' and refers the reader to Adler's book. These relations are the bridge from the Level-0 matrix dynamics to the claimed Level-I quantum gravity, so deferring them leaves the central claim unverifiable from the present text.
  3. [Section II, Eqs. (67)-(71)] The numerical 'predictions' are not independent tests of the theory. Eq. (71) for the proton mass, the identification of L_I with the Hubble scale, and the collapse rate L_I/c ∼ 10^{-17} s^{-1} all follow algebraically from the same assumed relation L_p^2 L_I = L^3 after fixing L to the nucleon Compton wavelength and L_I to cH_0^{-1}. The paper itself says 'If this is not a coincidence', signalling that the relation is fitted rather than derived. Presenting these as predictions obscures their origin.
  4. [Section II, Eq. (60)] The transition from the localised bosonic trace to the Einstein-Hilbert action uses the spectral-action formula ∑(λ_R^i)^2 ∝ ∫√g R. This formula is valid for the square of a Dirac-type operator in a heat-kernel expansion, but the manuscript does not show that the D_B obtained from the Level-0 dynamics is such an operator on a Riemannian manifold, nor that spontaneous localisation of the fermionic sector produces the eigenvalue distribution required by the trace formula. This is an additional unproven step between Eqs. (59) and (64).
minor comments (4)
  1. [Title and references] There are typographical errors that should be corrected: 'spontaneo us' appears in the title and 'qantum' appears in reference [1].
  2. [Section II, Eq. (69)] For a mass m = \hbar/(Lc), the standard Schwarzschild radius is R_S = 2 L_P^2/L (in units with G = L_P^2 and c = 1), whereas the text writes L_P^2/L^2 ≡ R_S/L; a factor of two is missing or a non-standard convention must be stated.
  3. [Section II, Eq. (58)] The notation L_I is introduced in Eq. (58) as the imaginary-part length, but its physical meaning is not stated until Eq. (63); a brief comment near Eq. (58) would improve readability.
  4. [Section II, Eq. (56)] The stochastic operator H(τ) is introduced by fiat; the paper should clarify whether this stochasticity is a consequence of the underlying anti-self-adjoint Hamiltonian or an additional assumption.

Circularity Check

2 steps flagged · score 7.0 of 10

The point-particle matter action in Eq. (63), and the subsequent 'predictions' of the collapse rate and proton mass, are rearrangements of the ad hoc relation L_p^2 L_I = L^3 introduced to make that action work; the Dirac/black-hole interpolation is the same relation restated.

  1. fitted input called prediction [Section II, between Eqs. (62) and (63), and after Eq. (66); Eq. (71)]
    "We make the assumption ... that spontaneous localisation localises the STM atom to a spatial volume L^3 such that L^2_p L_I = L^3. ... the contribution to the matter source action becomes ¯h ∫√g d^4x [L^{-2}_p × 1/L_I × 1/L] = mc ∫ ds. ... Let us return to our assumption LI = L^3/L^2_P, which as we will now see, has profound consequences. ... this relation allows us to 'predict' the mass m_P r of a proton ..."

    The relation L_p^2 L_I = L^3 is the only bridge converting the localised trace term into the relativistic point-particle action mc∫ds in Eq. (63). It is imposed, not derived from the spontaneous-localisation dynamics. The paper then presents consequences of this same relation as independent findings: with L the nucleon Compton wavelength it gives L_I ~ 10^27 cm and collapse rate L_I/c ~ 10^-17 s^-1, and Eq. (71) rearranges the same relation to 'predict' the proton mass. These are arithmetic reorganisations of the assumed input, so the matter action and the subsequent numerical 'predictions' reduce to the original assumption by construction.

  2. self definitional [Section II, Eqs. (67)-(69)]
    "[D_B + D_F]ψ = (1/L)(1 + iL^2_P/L^2)ψ ... define a complex length scale L_com by L_com/L^2 = (1/L)(1 + iL^2_P/L^2) ≡ (1/L)(1 + iR_S/L) =⇒ L_com = L + iR_S"

    Since mass was defined earlier as m = ¯h/(Lc), the Schwarzschild radius of that mass is R_S = 2L_p^2/L (the text uses the numerical factor 1). Substituting the assumed relation L_I = L^3/L_p^2 into the eigenvalue therefore makes the imaginary part equal to R_S by definition. The claimed interpolation between Dirac fermion and black hole, and the duality L ↔ L_p^2/L, are restatements of these definitions; the limit R_S ≫ L is just L ≪ L_p (m ≫ m_Pl), already built into the definitions of m and G. No independent dynamics selects these limits.

full rationale

The paper contains a genuine derivation chain for the trace-dynamics equations (Lagrange/Hamilton equations, Adler-Millard charge), but the central passage from Level 0/I to classical general relativity with matter is not self-contained. Eq. (60) uses the standard heat-kernel spectral action (cited to Landi and Connes), which is external mathematical input and does not by itself create circularity. The circularity enters in Eqs. (62)-(63): the relation L_p^2 L_I = L^3 is assumed ad hoc so that the localised matter trace term becomes mc∫ds, and then the same relation is used to 'predict' the Hubble-scale L_I, the 10^-17 s^-1 collapse rate, the proton mass, and the Dirac/black-hole unification of Eq. (68). These are not independent predictions; they are rearrangements of the imposed input. The paper explicitly labels the step an 'assumption', but that does not prevent it from being load-bearing and then being reused as a prediction. The claimed origin of spontaneous localisation is likewise an input (the anti-self-adjoint part is built into the choice of β1, β2), though the paper is transparent about this being part of the assumed Lagrangian. No separate self-citation load-bearing circularity is found, because the cited prior papers supply the framework rather than a result that is being relabeled as new. Overall, the matter-sector derivation and the headline 'predictions' reduce by construction to the assumed L_p^2 L_I = L^3 relation, giving score 7.

Assumptions & free parameters 4 free parameters · 8 assumptions · 4 invented entities

The central derivation rests on several postulates borrowed from noncommutative geometry and trace dynamics, plus an ad hoc relation LI = L^3/Lp^2 and unspecified beta matrices. The paper introduces multiple entities without independent falsifiable handles.

free parameters (4)
  • Length scale L of the STM atom = L = hbar/(mc) at Level I; unspecified at Level 0
    The STM atom is assigned an area L^2 at Level 0; later L is identified with the Compton wavelength, which makes the point-particle action (Eq. 63) possible.
  • Imaginary eigenvalue LI via Lp^2 LI = L^3 = LI = L^3/Lp^2; for a nucleon about 10^27 cm
    Introduced ad hoc after Eq. (62) to turn the localised trace term into the point-particle action; it also produces the collapse rate and proton mass predictions.
  • Action constant kappa/C0 = chosen to recover Einstein-Hilbert action
    The constant kappa in Eq. (1) is tuned so the heat kernel expansion gives the correct GR coefficient; this is a matching condition, not a prediction.
  • beta1 and beta2 = unspecified constant fermionic matrices
    They are chosen to keep the Lagrangian bosonic and to retain the fermionic kinetic term; their values are not determined by any equation in the paper.
assumptions (8)
  • standard math Grassmann graded trace derivative rules and adjointness properties (Eqs. 9-11)
    Background from Adler's trace dynamics, used without proof.
  • domain assumption The spectral action of non-commutative geometry, truncated at second order in the heat kernel expansion, equals the Einstein-Hilbert action (Eq. 60)
    Invoked to convert the localised sum of eigenvalues into integral sqrt(g) R; this is a known result from noncommutative geometry, but the truncation and coefficient matching are assumed.
  • domain assumption Trace dynamics statistical mechanics applies to this Lagrangian and yields quantum commutation relations (Eqs. 53-55)
    The paper does not carry out the statistical thermodynamics; it defers to Adler's book and a forthcoming paper.
  • domain assumption The anti-self-adjoint part of the Hamiltonian or Adler-Millard charge is the source of spontaneous localisation and can be represented by a stochastic function added to the Schrodinger equation (Eq. 56)
    Borrowed from trace dynamics and collapse models; the paper asserts it with no derivation for this specific Lagrangian.
  • ad hoc to paper The eigenvalues of the full Dirac operator D = DB + DF are complex numbers lambda = 1/L + i/L_I (Eq. 58)
    This spectral ansatz is stated, not derived from the action; it is used to define the length scales that produce the point-particle action.
  • domain assumption Norm preservation of the state vector under stochastic evolution is enforced by redefining the state vector as Psi divided by its norm
    The paper motivates this from free-particle geodesic motion, but it is the standard collapse-model prescription, added by hand.
  • ad hoc to paper beta1 and beta2 are constant fermionic matrices that neither both commute nor both anti-commute with qF_dot
    Needed to retain the fermionic kinetic term Tr[beta1 qF_dot beta2 qF_dot]; no physical principle determines them.
  • ad hoc to paper qB is self-adjoint, and beta1 and beta2 are self-adjoint (Eq. 47)
    Adopted for simplicity to make pB self-adjoint under condition (46).
invented entities (4)
  • STM atom (atom of space-time-matter)
    purpose: Fundamental quantum entity described by a trace action; its bosonic part gives gravity and its fermionic part gives matter, with an associated area L^2.
    No detection or independent handle is provided; it is the central postulate of the series.
  • qB and qF operators
    purpose: Bosonic and fermionic matrix degrees of freedom whose combinations form the Dirac operator D = DB + DF.
    The split q = qB + qF is assumed; no observable is defined outside the theory.
  • Constant fermionic matrices beta1 and beta2
    purpose: Make the trace Lagrangian bosonic and preserve the qF_dot qF_dot term; their anti-commutation properties control spontaneous localisation.
    No values or physical origin are given; the paper itself notes being 'compelled' to introduce two such matrices.
  • DF (fermionic Dirac operator)
    purpose: Fermionic part of the Dirac operator; its imaginary eigenvalue LI sets the collapse scale and is related to an asymmetric metric and torsion.
    DF's interpretation is speculative, with the paper saying 'our guess is...'; no independent evidence is provided.

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Pith. "Pith review of Proposal for a new quantum theory of gravity III: Equations for quantum gravity, and the origin of spontaneous localisation." pith.science (2026). https://pith.science/paper/BT7YQSX4

@misc{pith2026190804309,
  author       = {Pith},
  title        = {Pith review of: Proposal for a new quantum theory of gravity III: Equations for quantum gravity, and the origin of spontaneous localisation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BT7YQSX4}},
  note         = {Machine review of arXiv:1908.04309}
}
read the original abstract

We present a new, falsifiable, quantum theory of gravity, which we name Non-commutative Matter-Gravity. The commutative limit of the theory is classical general relativity. In the first two papers of this series, we have introduced the concept of an atom of space-time-matter [STM], which is described by the spectral action in non-commutative geometry, corresponding to a classical theory of gravity. We used the Connes time parameter, along with the spectral action, to incorporate gravity into trace dynamics. We then derived the spectral equation of motion for the gravity part of the STM atom, which turns out to be the Dirac equation on a non-commutative space. In the present work, we propose how to include the matter (fermionic) part and give a simple action principle for the STM atom. This leads to the equations for a quantum theory of gravity, and also to an explanation for the origin of spontaneous localisation from quantum gravity. We use spontaneous localisation to arrive at the action for classical general relativity [including matter sources] from the action for STM atoms.

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Forward citations

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Reference graph

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